L(s) = 1 | + (1.73 − i)2-s + (−2.54 + 4.40i)3-s + (1.99 − 3.46i)4-s + (9.68 + 5.59i)5-s + 10.1i·6-s + (−10.2 + 17.6i)7-s − 7.99i·8-s + (0.565 + 0.978i)9-s + 22.3·10-s + 15.2·11-s + (10.1 + 17.6i)12-s + (43.1 + 24.9i)13-s + 40.8i·14-s + (−49.2 + 28.4i)15-s + (−8 − 13.8i)16-s + (24.3 − 14.0i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.489 + 0.847i)3-s + (0.249 − 0.433i)4-s + (0.866 + 0.500i)5-s + 0.692i·6-s + (−0.551 + 0.954i)7-s − 0.353i·8-s + (0.0209 + 0.0362i)9-s + 0.707·10-s + 0.417·11-s + (0.244 + 0.423i)12-s + (0.920 + 0.531i)13-s + 0.779i·14-s + (−0.847 + 0.489i)15-s + (−0.125 − 0.216i)16-s + (0.347 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.774i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.74409 + 0.826817i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74409 + 0.826817i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.73 + i)T \) |
| 37 | \( 1 + (-224. + 15.1i)T \) |
good | 3 | \( 1 + (2.54 - 4.40i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (-9.68 - 5.59i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (10.2 - 17.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-43.1 - 24.9i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-24.3 + 14.0i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (68.6 + 39.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 119. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 162. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 119. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (57.4 - 99.4i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + 382. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (36.3 + 62.9i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (24.4 - 14.1i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-71.0 - 41.0i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-87.2 + 151. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (514. - 890. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 32.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + (461. + 266. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-590. - 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.31e3 + 756. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.75e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.18614957682922699976527364090, −13.18096933349074079481356786083, −11.95894377901576862936390431481, −10.85985361551298069620664589838, −10.00620783039098991623373891545, −8.943108921096314095047422630080, −6.49611496849524703876878948431, −5.69072951443446091031440877807, −4.20365603955815523861031068065, −2.41852608031923074921831684227,
1.28399509663510550758058348325, 3.81174215335531046426560663602, 5.71274709199986656646815979504, 6.48872205271282040147257888493, 7.70701842798012647462687616413, 9.362474741671031568599781811116, 10.73647757972315778200029664222, 12.12533992897764162737445618676, 13.19671089224111533231263261929, 13.41544447304793121237413039880